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\begin{document}
\maketitle
%---------------------------------------------------------------
\section*{Instructions}
\begin{itemize}
	\item From now on, feel free to use AI assistants for code generation,
		but we will check that you understand what your code is doing.
		We strongly advise to use the one provided by W\&M as we instructed
		you. {\bf Do not use AI for the report generation!} It is ok to ask AI
		for exotic parameters to improve your plots or check grammar but do not use AI to write the text of the
		report.
	\item Submit written report in PDF format (with inlined code snippets, if it is
	  needed or required). Report must be submitted to Gradescope \lstinline{hw03report}
		assignment. Remember that automatic checker test the
		functionality but not your test
		cases.
	\item Submit Python code to Gradescope \lstinline{hw03code} assignment, only for
		the problems which explicitly require it.
	\item When required, all relevant code for a given problem should be in
		file named as \lstinline{pN.py} where `N' stands for problem
		number, e.g. \lstinline{p1.py}, \lstinline{p2.py}, or
		\lstinline{p10.py}. All code files should be contained in one `zip'
		archive and uploaded for grading to Gradescope.
	\item The class web page will have templates with function definitions to which
		you must adhere!
	\item Do not forget to discuss test cases in your report.
	\item Review the function \lstinline{scipy.optimize.curve_fit}
		from \lstinline{scipy} package, and use it where the fitting required.
	\item If you asked to fit something, show in your report the plot
		of the data and the fitted curve.
	\item All data files are provided at the class web page and saved
		in comma separated format (CSV). In general, they can be
		loaded with
		\lstinline{data=np.loadtxt("filename.dat", delimiter=",", skiprows=Num)}
		command. Some of them have header, i.e. description of
		columns (than you need to use \lstinline{skiprows} argument,
		some do not have it. You can see if the header is present
		by opening a file in a text editor.
\end{itemize}

\problem{(5 points)}
%---------------------------------------------------------------
This problem requires code submission in the file \lstinline{p1.py}.

During photo processing quite often is needed to `smooth' or defocus a
photo. At the simplest level it can be done by creating a new 2D array
where every cell is averaged value of the cell itself
with it nearest neighbor as shown in~\ref{fig:smooth}. In this task we will
implement the periodic boundary condition where a neighbor could be on the
opposite end of the array (top and bottom, or left and right).

\begin{figure}[h] % 'h' stands for here, there are other specifiers too.
	\center
	\includegraphics[width=0.4\columnwidth,angle=0]{pics/array_averaging.pdf}
	\includegraphics[width=0.4\columnwidth,angle=0]{pics/array_border_periodic.pdf}
	\caption{
		\label{fig:smooth}
		Array smoothing. The smoothed location shown in blue and
		the neighbors as red. Left standard case with resulting
		value $(12+13+14+8+18)/5$, right shows
		the periodic boundary condition with resulting value
		$(2+3+4+8+23)/5$.
	}
\end{figure}

Implement a smoothing function \lstinline{smooth(dataIn)}  where
\lstinline{dataIn} is a 2D array of real numbers with arbitrary size (read about
how to use outcome of \lstinline{dataIn.shape} to get the array shape, i.e.
number of rows and columns). The function must return the smoothed array of
the same shape. The function must return the smoothed array of the same
shape. The function must return the smoothed array of the same shape.



\problem{(5 points)}
%---------------------------------------------------------------
Recall the problem about resistor from the previous homework 2. This time
we will use a fitting technique to extract resistance.

Download  data   file  \lstinline{'hw02dataset.dat'}  from  the   class  web page.  It
represents the result  of someone's attempt  to find  the resistance of a  sample via
measuring voltage  drop ($V$),  which is  the data in  the 1st  column, and
current ($I$), listed in the 2nd  column, passing through the resistor.
Judging  by the  number of  samples it  was an  automated
measurement.

Using Ohm's law $V=R I$  and a linear fit of the data  with one free parameter ($R$)
find the resistance ($R$) of this sample. What are the
errorbars/uncertainty of this estimate? Does it come close to the one which
you obtained via the method used in homework 2?

\problem{(5 points)}
%---------------------------------------------------------------
You are making a speed detector based on the Doppler effect. Your device detects
dependence of the signal strength vs. time, which is recorded in the
\lstinline{'hw_fit_cos_problem.dat'} file (the first column is time and the second is the
signal strength).

Fit the data with 
\begin{eqnarray*}
	A \cos(\omega t +\phi)
\end{eqnarray*}
where $A$,  $\omega$ and $\phi$ are the amplitude, the frequency and the phase of the signal, and $t$ is time.

Find fit parameters (the amplitude, the frequency and the phase of the
signal) and their uncertainties. 

\problem{(Bonus 2 points)}
This is for the physicists among us.
Provided that the above radar was using radio frequency, could you estimate
the velocity measurement uncertainty? Is it a good detector to measure a
car's velocity?


\problem{(5 points)}
%---------------------------------------------------------------
Experiment to do at home. Make a pendulum of variable length (0.1m, 0.2m,
0.3m, and so on up to 1m). Measure how many round trip (back and forth) swings the
pendulum with each particular length does in 20 seconds (clearly you will have to round to the nearest
integer).
Save your observations into the simple text file with comma  separated
values (commongly referred CSV) in
columns. The first column should be the length of the pendulum in meters,
the second column the number of full swings in 20 seconds.

Write a script which loads this data file, and extract acceleration due to
gravity ($g$) from the properly fitted experimental data.
Recall that the period of the oscillation of
a pendulum with the length $L$ is given by the following formula
\begin{eqnarray*}
	T = 2 \pi \sqrt{ \frac{L}{g} }
\end{eqnarray*}

\problem{(5 points)}
%---------------------------------------------------------------
In optics, the propagation of the laser beams is often described in the
Gaussian beams formalism. Among other things, it says that the optical beam
intensity cross section is described by the Gaussian profile (hence, the name of
the beams)
\begin{eqnarray*}
	I(x) = A \exp{\left( -\frac{(x-x_o)^2}{w^2} \right)} + B
\end{eqnarray*}
where $A$ is the amplitude, $x_o$ is the position of the maximum intensity,
$w$ is the characteristic width of the beam (width at $1/e$ intensity
level), and $B$ is the background illumination of the sensor.

Extract  the  $A$, $x_o$, $w$, and  $B$ with their uncertainties  from the real
experimental data  contained in the file  \lstinline{'gaussian_beam.dat'}, where the
first column is the position ($x$) in  meters and the second column is the beam
intensity in arbitrary units. 

Is the suggested model describe the experimental data well? Why so?

\problem{(5 points)}
Fit the data from the file \lstinline{'data_to_make_fit_choice.dat'} with the above Gaussian
profile. Is the resulting fit  good one or not? Why so? Compare it to the
Lorentzian model which is given by
\begin{eqnarray*}
	I(x) = A \frac{w^2}{(x-x_o)^2+{w^2}} + B
\end{eqnarray*}

To assist your decision make plot of the residuals in both cases.

\end{document}